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Next: 5. Acknowledgments Up: Comparison of relativistic partial Previous: 3.2 Comparison with the

   
4. Summary and future work

We have reported a comparison of results obtained using our relativistic partial wave and multipole calculation of the unpolarized triply differential bremsstrahlung cross section with predictions of the (Elwert-)Bethe-Heitler and Elwert-Haug approximations. Our results have been summarized in Table 7.1. For the high Z cases considered here and energies from 50-450 keV we find that there is no significant region of systematic agreement with the (Elwert-)Bethe-Heitler approximation for $d^3\!\sigma$. At small momentum transfers, $q<0.6(Z\alpha)\,mc$ for forward emission and with increasing momentum transfers for non-forward emission, the Bethe-Heitler results tend to overestimate the exact partial wave results. For larger momentum transfers they tend to underestimate the exact partial wave results. We see differences ranging from factors of two to factors of four, typically much larger than those seen in the $d^2\!\sigma$ results. The reduced differences in $d^2\!\sigma$ are attributed to cancellation of the differences in the small and large momentum transfer regions when integrating over $\theta_2$. This provides an example which confirms the expectation that observation of $d^3\!\sigma$ provides a more stringent test of the underlying model. In the case of $d^2\!\sigma$, experimental uncertainty of less than 10-20% is required to differentiate results consistent with Bethe-Heitler approximation, except in the tip region, (or 5-10% if the Elwert factor is included) from those consistent with the exact partial wave calculations. In the case of $d^3\!\sigma$ measurements can distinguish between the two theories even when experimental uncertainty is as large as 30%. While use of the Elwert factor typically improved agreement in $d^2\!\sigma$, we find that in $d^3\!\sigma$ it gives mixed results - causing larger differences at small momentum transfers but improving the agreement somewhat at larger momentum transfers.

In a similar comparison of the Elwert-Haug approximation with the exact partial wave results for the unpolarized cross section for high Z and energies from 50-450 keV, we find that at small momentum transfers the Elwert-Haug results also tend to overestimate the exact partial wave results. However, this overestimation is typically less than 20% for the cases considered here; quite good agreement was found in our calculations for E1=300 keV with $E_2\ge 150$ keV at low momentum transfers and small photon emission angles. We find that for small emission and scattering angles the Elwert-Haug approximation performs well at energies greater than 100 keV. For larger emission angles or larger momentum transfers the Elwert-Haug results tend to significantly underestimate the exact partial wave results, approaching the Elwert-Bethe-Heitler results. We find that the integrated cross section $d^2\!\sigma$ (as reported by Tseng [16] and Fink [18]) underestimates the exact partial wave results at most energies, reflecting the underestimate at large momentum transfers in $d^3\!\sigma$.

We now have available results, for coplanar and non-coplanar geometries, over a broad range of atomic species and electron energies, which we can compare with anticipated future experiments [15]. Future theoretical efforts could include examination of bremsstrahlung from electrons of lower or higher energies than considered here. Lower energy cases would be of particular interest in comparisons with predictions of so-called atomic or polarizational bremsstrahlung [19]. At higher energies one should see the nature of convergence toward high energy limit forms. We should also note that limitations in our current computer program have prevented us from considering cases where the photon energy was low compared to the incident electron energy ( k/T1 < 0.35). Extension of our code to include such cases could be of interest for comparisons with future experiments.


next up previous
Next: 5. Acknowledgments Up: Comparison of relativistic partial Previous: 3.2 Comparison with the
Eoin Carney
1999-06-14